Question About Representation of Brownian Motion

Question

In Stochastic Calculus with Financial Applications by J. Michael Steele he makes a specific representation of a Brownian Motion using wavelets. At one point he calculates the covariance, and he uses this equality: $E[\sum_{n=0}^\infty\lambda_n Z_n\Delta_n(s)\sum_{m=0}^\infty\lambda_mZ_m\Delta_m(t)]=\sum_{n=0}^\infty\lambda_n^2\Delta_n(s)\Delta_n(t)]$. I am just starting out in Real Analysis and am self taught so maybe I missed something elementary, but isn't he using Fubini's Theorem on that double sum? And if he is then when did he justify using it? To my knowledge in order to use $E(\sum\sum f)=\sum\sum E(f)$ one needs to show $E(\sum\sum |f|)<\infty$ and I do not see where he showed that.

Answer 1

$X_s=\sum_{n=0}^\infty\lambda_n Z_n \Delta_n(s)$ is absolutely convergent. So the product $X_sX_t$ of two absolutely convergent series is an absolutely convergent series. Thus you can rearrange the terms any way you like.